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<hr>
<h2 id="1-点特征直方图-PFH"><a href="#1-点特征直方图-PFH" class="headerlink" title="1. 点特征直方图 (PFH)"></a>1. 点特征直方图 (PFH)</h2><h3 id="1-1-基本思想"><a href="#1-1-基本思想" class="headerlink" title="1.1 基本思想"></a>1.1 基本思想</h3><p>PFH 通过分析一个点与其邻域点之间的<strong>局部几何关系</strong>，构建一个多维直方图来描述该点周围的曲面特性。它利用每对点之间的法向量夹角关系来捕捉局部表面的几何形状。</p>
<h3 id="1-2-计算流程"><a href="#1-2-计算流程" class="headerlink" title="1.2 计算流程"></a>1.2 计算流程</h3><ol>
<li><p><strong>法向量估计</strong><br>对点云中的每个点，先估计其单位法向量（通常使用主成分分析 PCA）。</p>
</li>
<li><p><strong>邻域选择</strong><br>对于点 $ p_i $，找出其在半径 $ r $ 内的所有邻近点 $ {p_j} $。</p>
</li>
<li><p><strong>计算特征值（角度关系）</strong><br>对于每对点 $ (p_i, p_j) $，构造一个局部坐标系，并计算以下四个角度特征：</p>
<ul>
<li>$ \alpha $：从 $ p_i $ 指向 $ p_j $ 的向量与 $ p_i $ 法向量的夹角。</li>
<li>$ \phi $：从 $ p_i $ 指向 $ p_j $ 的向量与 $ p_j $ 法向量的夹角。</li>
<li>$ \theta $：两个法向量在垂直于连接向量平面上的夹角。</li>
<li>$ d $：两点之间的欧氏距离（可选）。</li>
</ul>
<p>这些特征共同描述了局部曲面的几何属性。</p>
</li>
<li><p><strong>构建直方图</strong><br>将上述四个特征值量化并组合成一个多维直方图（例如，将每个维度划分为若干区间，最终形成 $ n $ 维直方图），归一化后作为该点的 PFH 描述子。</p>
</li>
</ol>
<h3 id="1-3-特点"><a href="#1-3-特点" class="headerlink" title="1.3 特点"></a>1.3 特点</h3><ul>
<li><strong>优点</strong>：<ul>
<li>对噪声和采样密度变化具有一定鲁棒性。</li>
<li>能有效描述局部曲面形状（如平面、球面、边缘等）。</li>
</ul>
</li>
<li><strong>缺点</strong>：<ul>
<li>计算复杂度高：对每个点需计算其邻域内所有点对之间的关系，时间复杂度为 $ O(k^2) $，其中 $ k $ 是邻域点数。</li>
<li>不适合大规模点云实时处理。</li>
</ul>
</li>
</ul>
<hr>
<h2 id="2-快速点特征直方图-FPFH"><a href="#2-快速点特征直方图-FPFH" class="headerlink" title="2. 快速点特征直方图 (FPFH)"></a>2. 快速点特征直方图 (FPFH)</h2><h3 id="2-1-提出背景"><a href="#2-1-提出背景" class="headerlink" title="2.1 提出背景"></a>2.1 提出背景</h3><p>由于 PFH 计算开销大，FPFH 被提出作为其简化和加速版本。FPFH 保留了 PFH 的表达能力，但显著降低了计算复杂度。</p>
<h3 id="2-2-核心思想"><a href="#2-2-核心思想" class="headerlink" title="2.2 核心思想"></a>2.2 核心思想</h3><p>FPFH 的关键在于：</p>
<ul>
<li><strong>简化计算过程</strong>：不再计算邻域内所有点对之间的完整关系。</li>
<li>引入 <strong>SPFH（Simplified PFH）</strong> 概念，仅计算中心点与每个邻域点之间的特征对。</li>
<li>最终描述子由 SPFH 与邻域加权组合而成。</li>
</ul>
<h3 id="2-3-计算步骤"><a href="#2-3-计算步骤" class="headerlink" title="2.3 计算步骤"></a>2.3 计算步骤</h3><ol>
<li><p><strong>计算 SPFH</strong><br>对于点 $ p_i $，计算其与每个邻域点 $ p_j $ 的 PFH 特征（即 $ \alpha, \phi, \theta, d $），得到一组简化的特征直方图（SPFH），不进行全连接组合。</p>
</li>
<li><p><strong>构建 FPFH</strong><br>对每个邻域点 $ p_j $，获取其 SPFH，并加权平均到 $ p_i $ 的最终描述子中：<br>$$<br>\text{FPFH}(p_i) &#x3D; \text{SPFH}(p_i) + \frac{1}{k} \sum_{j&#x3D;1}^{k} \text{SPFH}(p_j)<br>$$<br>其中权重可基于距离衰减。</p>
</li>
<li><p><strong>归一化</strong><br>对最终直方图进行归一化，以增强对光照、密度等变化的鲁棒性。</p>
</li>
</ol>
<h3 id="2-4-特点"><a href="#2-4-特点" class="headerlink" title="2.4 特点"></a>2.4 特点</h3><ul>
<li><strong>优点</strong>：<ul>
<li>时间复杂度降低至 $ O(k) $，适合大规模点云处理。</li>
<li>保留了 PFH 大部分判别能力。</li>
<li>广泛集成于 PCL（Point Cloud Library）中，如 <code>pcl::FPFHEstimation</code>。</li>
</ul>
</li>
<li><strong>缺点</strong>：<ul>
<li>相比 PFH，细节信息略有损失（因未计算所有点对）。</li>
</ul>
</li>
</ul>
<hr>
<h2 id="3-PFH-vs-FPFH-对比"><a href="#3-PFH-vs-FPFH-对比" class="headerlink" title="3. PFH vs FPFH 对比"></a>3. PFH vs FPFH 对比</h2><table>
<thead>
<tr>
<th>特性</th>
<th>PFH</th>
<th>FPFH</th>
</tr>
</thead>
<tbody><tr>
<td>计算复杂度</td>
<td>$ O(k^2) $</td>
<td>$ O(k) $</td>
</tr>
<tr>
<td>描述能力</td>
<td>更完整，包含所有点对关系</td>
<td>稍弱，但足够用于多数任务</td>
</tr>
<tr>
<td>实际应用</td>
<td>较少，因速度慢</td>
<td>广泛使用（如配准、识别）</td>
</tr>
<tr>
<td>实现库支持</td>
<td>PCL 支持</td>
<td>PCL 默认推荐</td>
</tr>
<tr>
<td>对噪声鲁棒性</td>
<td>高</td>
<td>高</td>
</tr>
</tbody></table>
<blockquote>
<p>✅ <strong>建议</strong>：在大多数实际应用中，推荐使用 <strong>FPFH</strong>，因其在速度与性能之间取得了良好平衡。</p>
</blockquote>
<hr>
<h2 id="4-应用场景"><a href="#4-应用场景" class="headerlink" title="4. 应用场景"></a>4. 应用场景</h2><ul>
<li><strong>点云配准（Registration）</strong>：作为特征对应点匹配的基础（如结合 SAC-IA 或 FPFH + RANSAC）。</li>
<li><strong>物体识别与分类</strong>：用于提取局部特征，输入到分类器（如 SVM、随机森林）。</li>
<li><strong>场景分割与理解</strong>：辅助区分不同几何结构区域（平面、柱面、角落等）。</li>
</ul>
<hr>
<h2 id="5-PCL-中的使用示例（C-）"><a href="#5-PCL-中的使用示例（C-）" class="headerlink" title="5. PCL 中的使用示例（C++）"></a>5. PCL 中的使用示例（C++）</h2><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;pcl/features/fpfh.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;pcl/features/pfh.h&gt;</span></span></span><br><span class="line"></span><br><span class="line"><span class="comment">// 法向量估计（前置步骤）</span></span><br><span class="line">pcl::NormalEstimation&lt;pcl::PointXYZ, pcl::Normal&gt; norm_est;</span><br><span class="line">norm_est.<span class="built_in">setInputCloud</span>(cloud);</span><br><span class="line">norm_est.<span class="built_in">setRadiusSearch</span>(<span class="number">0.03</span>);</span><br><span class="line">norm_est.<span class="built_in">compute</span>(*normals);</span><br><span class="line"></span><br><span class="line"><span class="comment">// FPFH 计算</span></span><br><span class="line">pcl::FPFHEstimation&lt;pcl::PointXYZ, pcl::Normal, pcl::FPFHSignature33&gt; fpfh;</span><br><span class="line">fpfh.<span class="built_in">setInputCloud</span>(cloud);</span><br><span class="line">fpfh.<span class="built_in">setInputNormals</span>(normals);</span><br><span class="line">fpfh.<span class="built_in">setSearchMethod</span>(tree);</span><br><span class="line">fpfh.<span class="built_in">setRadiusSearch</span>(<span class="number">0.05</span>);</span><br><span class="line">pcl::PointCloud&lt;pcl::FPFHSignature33&gt;::<span class="function">Ptr <span class="title">fpfh_descriptors</span><span class="params">(<span class="keyword">new</span> pcl::PointCloud&lt;pcl::FPFHSignature33&gt;)</span></span>;</span><br><span class="line">fpfh.<span class="built_in">compute</span>(*fpfh_descriptors);</span><br></pre></td></tr></table></figure>

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href="#%E7%82%B9%E7%89%B9%E5%BE%81%E7%9B%B4%E6%96%B9%E5%9B%BE-PFH-%E4%B8%8E%E5%BF%AB%E9%80%9F%E7%82%B9%E7%89%B9%E5%BE%81%E7%9B%B4%E6%96%B9%E5%9B%BE-FPFH"><span class="toc-number">1.</span> <span class="toc-text">点特征直方图 (PFH) 与快速点特征直方图 (FPFH)</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-%E7%82%B9%E7%89%B9%E5%BE%81%E7%9B%B4%E6%96%B9%E5%9B%BE-PFH"><span class="toc-number">1.1.</span> <span class="toc-text">1. 点特征直方图 (PFH)</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#1-1-%E5%9F%BA%E6%9C%AC%E6%80%9D%E6%83%B3"><span class="toc-number">1.1.1.</span> <span class="toc-text">1.1 基本思想</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#1-2-%E8%AE%A1%E7%AE%97%E6%B5%81%E7%A8%8B"><span class="toc-number">1.1.2.</span> <span class="toc-text">1.2 计算流程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#1-3-%E7%89%B9%E7%82%B9"><span class="toc-number">1.1.3.</span> <span class="toc-text">1.3 特点</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-%E5%BF%AB%E9%80%9F%E7%82%B9%E7%89%B9%E5%BE%81%E7%9B%B4%E6%96%B9%E5%9B%BE-FPFH"><span class="toc-number">1.2.</span> <span class="toc-text">2. 快速点特征直方图 (FPFH)</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#2-1-%E6%8F%90%E5%87%BA%E8%83%8C%E6%99%AF"><span class="toc-number">1.2.1.</span> <span class="toc-text">2.1 提出背景</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-2-%E6%A0%B8%E5%BF%83%E6%80%9D%E6%83%B3"><span class="toc-number">1.2.2.</span> <span class="toc-text">2.2 核心思想</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-3-%E8%AE%A1%E7%AE%97%E6%AD%A5%E9%AA%A4"><span class="toc-number">1.2.3.</span> <span class="toc-text">2.3 计算步骤</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#2-4-%E7%89%B9%E7%82%B9"><span class="toc-number">1.2.4.</span> <span class="toc-text">2.4 特点</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#3-PFH-vs-FPFH-%E5%AF%B9%E6%AF%94"><span class="toc-number">1.3.</span> <span class="toc-text">3. PFH vs FPFH 对比</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#4-%E5%BA%94%E7%94%A8%E5%9C%BA%E6%99%AF"><span class="toc-number">1.4.</span> <span class="toc-text">4. 应用场景</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#5-PCL-%E4%B8%AD%E7%9A%84%E4%BD%BF%E7%94%A8%E7%A4%BA%E4%BE%8B%EF%BC%88C-%EF%BC%89"><span class="toc-number">1.5.</span> <span class="toc-text">5. PCL 中的使用示例（C++）</span></a></li></ol></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="fas fa-history"></i><span>最新文章</span></div><div class="aside-list"><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2025/10/05/2025-10-05-lio_sam/" title="lio_sam">lio_sam</a><time datetime="2025-10-05T02:43:51.000Z" title="发表于 2025-10-05 10:43:51">2025-10-05</time></div></div><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2025/08/12/2025-08-12-%E4%BD%BF%E7%94%A8docker%E6%90%AD%E5%BB%BApytorch%E7%8E%AF%E5%A2%83/" title="使用docker搭建pytorch环境">使用docker搭建pytorch环境</a><time datetime="2025-08-12T12:43:33.000Z" title="发表于 2025-08-12 20:43:33">2025-08-12</time></div></div><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2025/08/05/2025-08-05-docker/" title="docker">docker</a><time datetime="2025-08-05T11:02:36.000Z" title="发表于 2025-08-05 19:02:36">2025-08-05</time></div></div><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2025/08/05/2025-08-05-%E8%BF%90%E5%8A%A8%E5%AF%B9%E8%B1%A1%E5%88%86%E5%89%B2%E4%B8%8E%E9%85%8D%E5%87%86%E7%AE%97%E6%B3%95/" title="运动对象分割与配准算法">运动对象分割与配准算法</a><time datetime="2025-08-05T08:16:02.000Z" title="发表于 2025-08-05 16:16:02">2025-08-05</time></div></div><div class="aside-list-item no-cover"><div class="content"><a class="title" href="/2025/08/05/2025-08-05-%E6%9D%A1%E4%BB%B6%E6%AC%A7%E5%BC%8F%E8%81%9A%E7%B1%BB%E5%88%86%E5%89%B2/" title="条件欧式聚类分割">条件欧式聚类分割</a><time datetime="2025-08-05T07:56:39.000Z" title="发表于 2025-08-05 15:56:39">2025-08-05</time></div></div></div></div></div></div></main><footer id="footer"><div class="footer-other"><div class="footer-copyright"><span class="copyright">&copy;&nbsp;2025 By Fi9zero</span><span class="framework-info"><span>框架 </span><a target="_blank" rel="noopener" href="https://hexo.io">Hexo 7.3.0</a><span class="footer-separator">|</span><span>主题 </span><a target="_blank" rel="noopener" href="https://github.com/jerryc127/hexo-theme-butterfly">Butterfly 5.4.2</a></span></div><div class="footer_custom_text">Only light can attract bugs.</div></div></footer></div><div id="rightside"><div id="rightside-config-hide"><button id="readmode" type="button" title="阅读模式"><i class="fas fa-book-open"></i></button><button id="darkmode" type="button" title="日间和夜间模式切换"><i class="fas fa-adjust"></i></button><button id="hide-aside-btn" type="button" title="单栏和双栏切换"><i class="fas fa-arrows-alt-h"></i></button></div><div id="rightside-config-show"><button id="rightside-config" type="button" title="设置"><i class="fas fa-cog fa-spin"></i></button><button class="close" id="mobile-toc-button" type="button" title="目录"><i class="fas fa-list-ul"></i></button><button id="go-up" type="button" title="回到顶部"><span class="scroll-percent"></span><i class="fas fa-arrow-up"></i></button></div></div><div><script src="/js/utils.js"></script><script src="/js/main.js"></script><div class="js-pjax"><script>(() => {
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